How to Make It
Heat 2 tablespoons oil in a large Dutch oven over medium-high. Add chicken pieces; cook until browned, about 5 minutes per side.
Increase heat to high; add garlic and ginger, and cook, stirring constantly, until just fragrant, about 30 seconds. Add soy sauce; cook until almost completely reduced, about 1 minute. Stir in stock, vinegar, oyster sauce, bay leaves, berbere, and pepper; bring to a boil. Reduce heat to low; cover and simmer 45 minutes.
Remove Dutch oven from heat; let cool slightly, about 15 minutes. Remove chicken; set aside. Skim fat off liquid; discard. Bring liquid to a boil over medium; boil until reduced down to a glaze consistency, about 30 minutes. (You should have about 1/2 cup glaze.) Set aside.
Heat remaining 2 tablespoons oil in a large nonstick skillet over high. Add eggplant; cook until browned on all sides, 8 to 10 minutes. Add glaze mixture to skillet; toss to coat eggplant. Transfer to serving platter. Garnish with basil; serve with chicken, or reserve chicken for another use.